Let G1 be a planar graph such that all cycles of length at most 4 are independent and let G2 be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of G1 and G2 can be equitably partitioned into t subsets for every t ≥ 3 so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang

Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. In general graphs, the ratio cfvs(G)/fvs(G) can be arbitrarily large. We study the interdependence between fvs(G) and cfvs(G) in graph classes defined by excluding one induced subgraph H. We show that the ratio cfvs(G)/fvs(G) is bounded by a co...

In graph sparsification, the goal has almost always been of global nature: compress a graph into a smaller subgraph (sparsifier) that maintains certain features of the original graph. Algorithms can then run on the sparsifier, which in many cases leads to improvements in the overall runtime and memory. This paper studies sparsifiers that have bounded (maximum) degree, and are thus locally spars...

Let R be the set of real numbers and D be a subset of the positive real numbers. The distance graph G(R, D) is a graph with the vertex set R and two vertices x and y are adjacent if and only if |x − y| ∈ D. In this work, the vertex arboricity (i.e., the minimum number of subsets into which the vertex set V (G) can be partitioned so that each subset induces an acyclic subgraph) of G(R, D) is det...

Using a fixed set of colors C, Ann and Ben color the edges of a graph G so that no monochromatic cycle may appear. Ann wins if all edges of G have been colored, while Ben wins if completing a coloring is not possible. The minimum size of C for which Ann has a winning strategy is called the game arboricity of G, denoted by Ag(G). We prove that Ag(G) ≤ 3k for any graph G of arboricity k, and that...

We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(|E(G)|/ε) time algorithm which finds an orientation of an input graph G with outdegree at most d(1 + ε)d∗e, where d∗ is the maximum density of a subgraph of G. It is known that the optimal v...

In this paper, we present a new degree-based estimator for the size of maximum matching in bounded arboricity graphs. When graph is by $$\alpha $$ , gives +2$$ factor approximation size. For planar graphs, show does better and returns 3.5 Using estimator, get results approximating graphs streaming distributed models computation. particular, vertex-arrival streams, randomized $$O\left( \frac{\sq...